Integrating Multiple-point Statistics into Sequential Simulation Algorithms

Abstract

Most conventional simulation techniques account for two-point statistics via the modeling of the variogram of the regionalized variable or of its indicators. These techniques cannot control the reproduction of multiple-point statistics or higher order features that may be critical for the performance of the models given the goal at hand (flow simulation in petroleum applications, planning and scheduling for mining applications).Multiple-point simulation is a way to deal with this situation. It has been implemented for categorical variables, yet the demand of large data sets (training images) to infer the multiple- point statistics has impeded its use in the case of continuous variables. The main problem is that multiple-point statistics are characterized by discretizing the continuous variable by a set of thresholds and coding them as indicators, which results in a loss of resolution between thresholds.We propose a method to incorporate multiple-point statistics into sequential simulation of continuous variables. Any sequential algorithm can be used. The method proceeds as follows. First, the multiple-point statistics are inferred from a training data set or training image with the typical indicator approach. The conditional probabilities given multiple-points data events enable to update the conditional distributions obtained by the sequential algorithm that uses the conventional two-point statistics. The key aspect is to preserve the shape of the conditional distribution between thresholds after updating the probability for the cutoffs used to infer the multiple-point statistics. Updating takes place under the assumption of conditional independence between the conditional probability obtained from the training set and the one retrieved from the conditional probability defined by the sequential method. The algorithm is presented for any sequential algorithm and then illustrated on a real data set using the sequential indicator and Gaussian simulation methods. The advantages and drawbacks of this proposal are pointed out.